Index Theory, Gerbes, and Hamiltonian Quantization

TL;DR

This paper develops an index theory approach to construct fermionic Fock space bundles parametrized by vector potentials, connecting it with gerbe theory and providing a unified framework for gauge and gravitational fields.

Contribution

It introduces an Atiyah-Patodi-Singer index construction for fermionic bundles, linking it with gerbe theory and extending applicability to gravitational backgrounds.

Findings

Explicit computation of the Dixmier-Douady class.

Unified treatment of gauge and gravitational cases.

Simplified derivation of Schwinger terms.

Abstract

We give an Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parametrized by vector potentials in odd space dimensions and prove that this leads in a simple manner to the known Schwinger terms (Faddeev-Mickelsson cocycle) for the gauge group action. We relate the APS construction to the bundle gerbe approach discussed recently by Carey and Murray, including an explicit computation of the Dixmier-Douady class. An advantage of our method is that it can be applied whenever one has a form of the APS theorem at hand, as in the case of fermions in an external gravitational field.